Precalculus Examples

$(2 x - 3) (x + 2) (x - 3) \leq 0$

Step 1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 x - 3 = 0$

$x + 2 = 0$

$x - 3 = 0$

Step 2

Set $2 x - 3$ equal to $0$ and solve for $x$ .

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Step 2.1

Set $2 x - 3$ equal to $0$ .

$2 x - 3 = 0$

Step 2.2

Solve $2 x - 3 = 0$ for $x$ .

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Step 2.2.1

Add $3$ to both sides of the equation.

$2 x = 3$

Step 2.2.2

Divide each term in $2 x = 3$ by $2$ and simplify.

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Step 2.2.2.1

Divide each term in $2 x = 3$ by $2$ .

$\frac{2 x}{2} = \frac{3}{2}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{3}{2}$

Step 2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{2}$

Step 3

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 3.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 4

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 4.2

Add $3$ to both sides of the equation.

$x = 3$

Step 5

The final solution is all the values that make $(2 x - 3) (x + 2) (x - 3) \leq 0$ true.

$x = \frac{3}{2}, - 2, 3$

Step 6

Use each root to create test intervals.

$x < - 2$

$- 2 < x < \frac{3}{2}$

$\frac{3}{2} < x < 3$

$x > 3$

Step 7

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 7.1

Test a value on the interval $x < - 2$ to see if it makes the inequality true.

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Step 7.1.1

Choose a value on the interval $x < - 2$ and see if this value makes the original inequality true.

$x = - 4$

Step 7.1.2

Replace $x$ with $- 4$ in the original inequality.

$(2 (- 4) - 3) ((- 4) + 2) ((- 4) - 3) \leq 0$

Step 7.1.3

The left side $- 154$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 7.2

Test a value on the interval $- 2 < x < \frac{3}{2}$ to see if it makes the inequality true.

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Step 7.2.1

Choose a value on the interval $- 2 < x < \frac{3}{2}$ and see if this value makes the original inequality true.

$x = 0$

Step 7.2.2

Replace $x$ with $0$ in the original inequality.

$(2 (0) - 3) ((0) + 2) ((0) - 3) \leq 0$

Step 7.2.3

The left side $18$ is greater than the right side $0$ , which means that the given statement is false.

False

Step 7.3

Test a value on the interval $\frac{3}{2} < x < 3$ to see if it makes the inequality true.

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Step 7.3.1

Choose a value on the interval $\frac{3}{2} < x < 3$ and see if this value makes the original inequality true.

$x = 2$

Step 7.3.2

Replace $x$ with $2$ in the original inequality.

$(2 (2) - 3) ((2) + 2) ((2) - 3) \leq 0$

Step 7.3.3

The left side $- 4$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 7.4

Test a value on the interval $x > 3$ to see if it makes the inequality true.

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Step 7.4.1

Choose a value on the interval $x > 3$ and see if this value makes the original inequality true.

$x = 6$

Step 7.4.2

Replace $x$ with $6$ in the original inequality.

$(2 (6) - 3) ((6) + 2) ((6) - 3) \leq 0$

Step 7.4.3

The left side $216$ is greater than the right side $0$ , which means that the given statement is false.

False

Step 7.5

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 2$ True

$- 2 < x < \frac{3}{2}$ False

$\frac{3}{2} < x < 3$ True

$x > 3$ False

$x < - 2$ True

$- 2 < x < \frac{3}{2}$ False

$\frac{3}{2} < x < 3$ True

$x > 3$ False

Step 8

The solution consists of all of the true intervals.

$x \leq - 2$ or $\frac{3}{2} \leq x \leq 3$

Step 9

Convert the inequality to interval notation.

$(- \infty, - 2] \cup [\frac{3}{2}, 3]$

Step 10